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# Program to Compute /n/Epsilon(I=1) Tau(n)

IP.com Disclosure Number: IPCOM000073456D
Original Publication Date: 1970-Dec-01
Included in the Prior Art Database: 2005-Feb-22
Document File: 2 page(s) / 38K

IBM

## Related People

Martino, MJ: AUTHOR

## Abstract

The left hand part of the drawing shows a routine for computing /n/Epsilon(I=1) Tau(n). Tau(n) is the number of integer divisors of n. Thus, Tau(6) = 4 because 6 has 4 divisors: 1, 2, 3 and 6. Similarly /n/epsilon(I=1)Tau(n) is the number of divisors for n and for each positive integer less than n. For example Tau(4) = 8 since the divisors of 4 are 1, 2 and 4; the divisors of 3 are 1 and 3; the divisors of 2 are 2 and 1; and the divisor of 1 is 1.

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Program to Compute /n/Epsilon(I=1) Tau(n)

The left hand part of the drawing shows a routine for computing /n/Epsilon(I=1) Tau(n). Tau(n) is the number of integer divisors of n. Thus, Tau(6) = 4 because 6 has 4 divisors: 1, 2, 3 and 6. Similarly /n/epsilon(I=1)Tau(n) is the number of divisors for n and for each positive integer less than n. For example Tau(4) = 8 since the divisors of 4 are 1, 2 and 4; the divisors of 3 are 1 and 3; the divisors of 2 are 2 and 1; and the divisor of 1 is 1.

In the initialize step, a storage or register location designated COFN is set to 1, a location I is set to 1, and a location TEMPA, which will hold intermediate results, is set to 0. Next, the number for which the Tau function is to be computed is located in a location designated n. In computing terms that may be divisors of n, it is unnecessary to go beyond the highest integer whose square is equal to or greater than n. The term is computed by a routine, not shown, and put in a location SQRTN.

From these values, the function c(n) is computed according to the routine shown to the right in the drawing. C(n) is the number of products, ab, that satisfy the relationship 0 </- a </- b </- n. Thus, C(4) = 5 because there are 5 products that meet the condition: lxl, 1x2, 1x3, 1x4 and 2x2. This function can be represented by a row and column arrangement of products with 1x1, 1x2...1xn in the left most column, the products 2x1, 2x2...2xn/2 in the next column and so on.

As the drawing sho...